6» Mr maxwell, ON FARADAY'S LINES OF FORCE. 



where ai b^ Cj, ui jSj 71 are any functions whatsoever, is capable of transformation into 



in which the quantities are found from the equations 



da, db, dcj 

 -—- + -—+ — -+ 47rpi = 0, 

 ax dy dz 



da, d&x dy, , 



-J- +-r + :T^+47rp/= 0; 

 d.v dy dz 



«oi3c7o^ are determined from a, 6, c, by the last theorem, so that 



dz dy dot 

 a-i 62 C2 are found from a, /3i 71 by the equations 



ddx dvi o 



a.,= — &c., 



J dx dy 



and p is found from the equation 



dJ'p d-p d^p , 



doe' dy' dz' "' 



For, if we put a^ in the form 



dA)_^o dV 

 dz dy dm 



and treat b^ and Cj similarly, then we have by integration by parts thA)ugh infinity, remem- 

 bering that all the functions vanish at the limits, 



or Q= + flf\ (47r Vp) - (a^a^ + fiA + 70C2) } djcdydx, 

 and by Theorem III, 



fffVp'dxdydz = fjfppdjsdydz, 

 so that finally 



Q = ffll'^'^Pp - ("o«2 + ^0^2 = y^2)]da>dydx. 



If Ci 61 Cj represent the components of magnetic quantity, and ai /3i 71 those of magnetic 

 intensity, then p will represent the real magnetic density, and p the magnetic potential or 

 tension. 02 62 <^2 '"'iH be the components of quantity of electric currents, and Oq /Sq 7o will be three 

 functions deduced from OiftjCi, which will be found to be the mathematical expression for 

 Faraday's Electro-tonic state. 



Let us now consider the bearing of these analytical theorems on the theory of magnetism. 

 Whenever we deal with quantities relating to magnetism, we shall distinguish them by the 

 suffix ( 1 ). Thus Oi 61 Cj are the components resolved in the directions of x, y, z of the 



